Weibull roots

The investigation of failures of manufactured components and systems, especially in the electronics and aerospace industries, has generated a variety of statistical models on which data analysis may be based. Each model uses a specific distibution of failure probabilities, and it is important to select a model that matches the actual distribution inherent in the product concerned. In the case of dielectric breakdown, where a large number of quite different modes of failure are known to occur, sometimes even together, the application of a particular statistical failure model must be approached with great caution. Nevertheless, one treatment, based on a Weibull distribution of failure probability, has taken root, and is most generally used in practice. For a dielectric, the Weibull failure probability function has the form... 

This is applicable to various carrier symmetries such as planar, cylindrical and spherical. Here Q is the amount of molecules released per unit exposed area of the carrier, t denotes time and a, b and k are constants. This power-law function is related to the Weibull function that has been suggested as a universal tool for describing release from both Euclidian and fractal systems, and may be considered as a short-time approximation of the latter (Kosmidis et al. 2003). The constant a takes initial delay and burst effects into account, and is a kinetic constant (Jamzad et al. 2005). The power law exponent, k, also called the transport coefficient, characterises the diffusion process and equals 0.5 for ordinary case I (or carrier conttoUed) diffusion in systems for which no swelling of the carrier material occurs, which can be expected for mesoporous material (Ritger and Peppas 1987). Diffusion-controlled release from a planar system, in which the carrier structure is inert, may be described by the Higuchi square-root-of-time law ... 

It may seem surprising that the notch support factor of brittle materials, like cast iron, is rather large (see figure 10.36). This is due to the fact that cracks in cast iron start at the graphite particles which act as inner defects and are statistically distributed. It is rather improbable that the crack that determines failure behaviour (the largest crack) is situated exactly at the notch root where the stress concentration becomes important. This is analogous to the dependence of the failure probability on the material volume according to the Weibull statistics (see section 7.3). This notch support in brittle materials also occurs under static loads. 

Fig. 4.16 Dependency of the Weibull exponent of a medium strength steel on notch root radius [8]...

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